Is a Rational Rotation Algebra a Cutdown of a Matrix Algebra?
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Let $theta=m/n$ and let $A_{theta}$ be the rational rotation C$^{*}$-algebra with rotation angle $theta$. I.e., $A_{theta}=C^{*}(u,v)$, where $u$ and $v$ are unitaries such that $vu=e^{2pi i theta}uv$. I know from here that $A_{theta}$ is not a full matrix algebra. Given that $A_{theta}$ has irreducible representations only of degree $n$, is it true that $A_{theta}$ is a cutdown of a full matrix algebra? I.e.,
Is there a space $C(X,M_{n}(mathbb{C}))$ and a projection $pin C(X,M_{n}(mathbb{C}))$, such that $A_{theta}$ is isomorphic to $pC(X,M_{n}(mathbb{C}))p$?
functional-analysis operator-theory operator-algebras c-star-algebras
asked Jan 14 at 13:34
ervxervx
10.3k31338
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add a comment |
$begingroup$
Let $theta=m/n$ and let $A_{theta}$ be the rational rotation C$^{*}$-algebra with rotation angle $theta$. I.e., $A_{theta}=C^{*}(u,v)$, where $u$ and $v$ are unitaries such that $vu=e^{2pi i theta}uv$. I know from here that $A_{theta}$ is not a full matrix algebra. Given that $A_{theta}$ has irreducible representations only of degree $n$, is it true that $A_{theta}$ is a cutdown of a full matrix algebra? I.e.,
Is there a space $C(X,M_{n}(mathbb{C}))$ and a projection $pin C(X,M_{n}(mathbb{C}))$, such that $A_{theta}$ is isomorphic to $pC(X,M_{n}(mathbb{C}))p$?
functional-analysis operator-theory operator-algebras c-star-algebras
asked Jan 14 at 13:34
ervxervx
10.3k31338
$endgroup$
add a comment |
$begingroup$
Let $theta=m/n$ and let $A_{theta}$ be the rational rotation C$^{*}$-algebra with rotation angle $theta$. I.e., $A_{theta}=C^{*}(u,v)$, where $u$ and $v$ are unitaries such that $vu=e^{2pi i theta}uv$. I know from here that $A_{theta}$ is not a full matrix algebra. Given that $A_{theta}$ has irreducible representations only of degree $n$, is it true that $A_{theta}$ is a cutdown of a full matrix algebra? I.e.,
Is there a space $C(X,M_{n}(mathbb{C}))$ and a projection $pin C(X,M_{n}(mathbb{C}))$, such that $A_{theta}$ is isomorphic to $pC(X,M_{n}(mathbb{C}))p$?
functional-analysis operator-theory operator-algebras c-star-algebras
asked Jan 14 at 13:34
ervxervx
10.3k31338
$endgroup$
Let $theta=m/n$ and let $A_{theta}$ be the rational rotation C$^{*}$-algebra with rotation angle $theta$. I.e., $A_{theta}=C^{*}(u,v)$, where $u$ and $v$ are unitaries such that $vu=e^{2pi i theta}uv$. I know from here that $A_{theta}$ is not a full matrix algebra. Given that $A_{theta}$ has irreducible representations only of degree $n$, is it true that $A_{theta}$ is a cutdown of a full matrix algebra? I.e.,
Is there a space $C(X,M_{n}(mathbb{C}))$ and a projection $pin C(X,M_{n}(mathbb{C}))$, such that $A_{theta}$ is isomorphic to $pC(X,M_{n}(mathbb{C}))p$?
functional-analysis operator-theory operator-algebras c-star-algebras
functional-analysis operator-theory operator-algebras c-star-algebras
asked Jan 14 at 13:34
ervxervx
10.3k31338
asked Jan 14 at 13:34
ervxervx
10.3k31338
asked Jan 14 at 13:34
ervxervx
10.3k31338
asked Jan 14 at 13:34
ervxervx
10.3k31338
asked Jan 14 at 13:34
ervxervx
10.3k31338
10.3k31338
add a comment |
add a comment |
1 Answer
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Yes, this is true.
As $theta$ is rational, it follows that $A_{theta}$ is Morita equivalent to $C(mathbb{T}^2)$, see for example The classification of rational rotation algebras- Brabanter.
However, two unital $C^*$-algebras are Morita equivalent if and only if each is a full corner of $ntimes n$ matrices over the other, for some $n$, see $C^*$-algebras associated with irrational rotations- Rieffel, Proposition 2.1.
Thus $A_{m/n}cong pM_k(C(mathbb{T}^2))p$, for some $kin mathbb{N}$ and full prjection $pin M_k(C(mathbb{T}^2))$.
answered Jan 19 at 13:38
Shirly GeffenShirly Geffen
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1 Answer
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1 Answer
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$begingroup$
Yes, this is true.
As $theta$ is rational, it follows that $A_{theta}$ is Morita equivalent to $C(mathbb{T}^2)$, see for example The classification of rational rotation algebras- Brabanter.
However, two unital $C^*$-algebras are Morita equivalent if and only if each is a full corner of $ntimes n$ matrices over the other, for some $n$, see $C^*$-algebras associated with irrational rotations- Rieffel, Proposition 2.1.
Thus $A_{m/n}cong pM_k(C(mathbb{T}^2))p$, for some $kin mathbb{N}$ and full prjection $pin M_k(C(mathbb{T}^2))$.
answered Jan 19 at 13:38
Shirly GeffenShirly Geffen
1,316614
$endgroup$
add a comment |
$begingroup$
Yes, this is true.
As $theta$ is rational, it follows that $A_{theta}$ is Morita equivalent to $C(mathbb{T}^2)$, see for example The classification of rational rotation algebras- Brabanter.
However, two unital $C^*$-algebras are Morita equivalent if and only if each is a full corner of $ntimes n$ matrices over the other, for some $n$, see $C^*$-algebras associated with irrational rotations- Rieffel, Proposition 2.1.
Thus $A_{m/n}cong pM_k(C(mathbb{T}^2))p$, for some $kin mathbb{N}$ and full prjection $pin M_k(C(mathbb{T}^2))$.
answered Jan 19 at 13:38
Shirly GeffenShirly Geffen
1,316614
$endgroup$
add a comment |
$begingroup$
Yes, this is true.
As $theta$ is rational, it follows that $A_{theta}$ is Morita equivalent to $C(mathbb{T}^2)$, see for example The classification of rational rotation algebras- Brabanter.
However, two unital $C^*$-algebras are Morita equivalent if and only if each is a full corner of $ntimes n$ matrices over the other, for some $n$, see $C^*$-algebras associated with irrational rotations- Rieffel, Proposition 2.1.
Thus $A_{m/n}cong pM_k(C(mathbb{T}^2))p$, for some $kin mathbb{N}$ and full prjection $pin M_k(C(mathbb{T}^2))$.
answered Jan 19 at 13:38
Shirly GeffenShirly Geffen
1,316614
$endgroup$
Yes, this is true.
As $theta$ is rational, it follows that $A_{theta}$ is Morita equivalent to $C(mathbb{T}^2)$, see for example The classification of rational rotation algebras- Brabanter.
However, two unital $C^*$-algebras are Morita equivalent if and only if each is a full corner of $ntimes n$ matrices over the other, for some $n$, see $C^*$-algebras associated with irrational rotations- Rieffel, Proposition 2.1.
Thus $A_{m/n}cong pM_k(C(mathbb{T}^2))p$, for some $kin mathbb{N}$ and full prjection $pin M_k(C(mathbb{T}^2))$.
answered Jan 19 at 13:38
Shirly GeffenShirly Geffen
1,316614
answered Jan 19 at 13:38
Shirly GeffenShirly Geffen
1,316614
answered Jan 19 at 13:38
Shirly GeffenShirly Geffen
1,316614
answered Jan 19 at 13:38
Shirly GeffenShirly Geffen
1,316614
1,316614
add a comment |
add a comment |
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